相关论文: Learning phase field mean curvature flows with neu…

We investigate a new phase field model for representing non-oriented interfaces, approximating their area and simulating their area-minimizing flow. Our contribution is related to the approach proposed in arXiv:2105.09627 that involves ad…

This paper considers and proposes some algorithms to compute the mean curvature flow under topological changes. Instead of solving the fully nonlinear partial differential equations based on the level set approach, we propose some…

We propose a deep learning strategy to estimate the mean curvature of two-dimensional implicit interfaces in the level-set method. Our approach is based on fitting feed-forward neural networks to synthetic data sets constructed from…

Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow…

The structure of many multiphase systems is governed by an energy that penalizes the area of interfaces between phases weighted by surface tension coefficients. However, interface evolution laws depend also on interface mobility…

This paper addresses the approximation of the mean curvature flow of thin structures for which classical phase field methods are not suitable. By thin structures, we mean surfaces that are not domain boundaries, typically higher codimension…

We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling…

In this paper, we explore the evolution of plane curves and surfaces governed by the hyperbolic mean curvature flow. We propose a mesh-free approach based on the physics-informed neural networks (PINNs), which eliminates the need for…

In this work, physics-informed neural networks are applied to incompressible two-phase flow problems. We investigate the forward problem, where the governing equations are solved from initial and boundary conditions, as well as the inverse…

The $L^2$ gradient flow of the Ginzburg-Landau free energy functional leads to the Allen Cahn equation that is widely used for modeling phase separation. Machine learning methods for solving the Allen-Cahn equation in its strong form suffer…

This paper describes a study based on computational fluid dynamics (CFD) and deep neural networks that focusing on predicting the flow field in differently distorted U-shaped pipes. The main motivation of this work was to get an insight…

An approximation model based on convolutional neural networks (CNNs) is proposed for flow field predictions. The CNN is used to predict the velocity and pressure field in unseen flow conditions and geometries given the pixelated shape of…

Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation…

The interaction of neural networks with physical equations offers a wide range of applications. We provide a method which enables a neural network to transform objects subject to given physical constraints. Therefore an U-Net architecture…

We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two--dimensional…

Phase-field models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial…

We present a novel deep learning framework for flow field predictions in irregular domains when the solution is a function of the geometry of either the domain or objects inside the domain. Grid vertices in a computational fluid dynamics…

We extend thresholding methods for numerical realization of mean curvature flow on obstacles to the anisotropic setting where interfacial energy depends on the orientation of the interface. This type of schemes treats the interface…

In this paper we investigate the numerical approximation of a variant of the mean curvature flow. We consider the evolution of hypersurfaces with normal speed given by $H^k$, $k \ge 1$, where $H$ denotes the mean curvature. We use a level…

Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature.…